We present a method for using aligned ontologies to merge taxonomically organized data sets that have apparently compatible schemas, but potentially different semantics for corresponding domains. We restrict the relationships involved in the alignment to basic set relations and disjunctions of these relations. A merged data set combines the domains of the source data set attributes, conforms to the observations reported in both data sets, and minimizes uncertainty introduced by ontology alignments. We find that even in very simple cases, merging data sets under this scenario is non-trivial. Reducing uncertainty introducced by the ontology alignments in combination with the data set observations often results in many possible merged data sets, which are managed using a possible worlds semantics. The primary contributions of this paper are a framework for representing aligned data sets and algorithms for merging data sets that report the presence and absence of taxonomically organized entities, including an efficient algorithm for a common data set merging scenario.